Physical Chemistry Third Edition

17.2 The Relative Schrödinger Equation. Angular Momentum 729

EXAMPLE17.1

The electron mass is 9. 10939 × 10 −^31 kg and the proton mass is 1. 672623 × 10 −^27 kg.

Calculate the ratio of the reduced mass of the hydrogen atom to the mass of the electron.

Solution

μ

memp

me(me+mp)



(1. 672623 × 10 −^27 kg)(9. 10939 × 10 −^31 kg)

1. 672623 × 10 −^27 kg+ 9. 10939 × 10 −^31 kg

 9. 10443 × 10 −^31 kg

μ

me



9. 10443 × 10 −^31 kg

9. 10939 × 10 −^31 kg

 0. 99946

PROBLEMS

Section 17.1: The Hydrogen Atom and the Central Force
System

17.1 The electron mass is 9. 10939 × 10 −^31 kg and the proton
mass is 1. 672623 × 10 −^27 kg. For a hydrogen atom with
the electron at a distance 1. 000 × 10 −^10 m from the
nucleus, find the distance from the center of mass to the
nucleus and to the electron. The calculation is easier if you
assume that the particles are temporarily on thexaxis with
the nucleus at the origin.

17.2 The mass of the earth is 5. 98 × 1024 kg and the mass of
the sun is 1. 99 × 1030 kg. Find the reduced mass of the
earth and the sun and express it as a percentage of the mass
of the earth.

17.3 The mass of the earth is 5. 98 × 1024 kg and the mass of

the sun is 1. 99 × 1030 kg. Assume that the earth is

1. 4957 × 108 km from the sun and find the distance from

the center of the sun to the center of mass of the earth and

the sun.

17.4 a.The mass of the moon is 7. 34 × 1022 kg and the mass

of the earth is 5. 98 × 1024 kg. Assume that the moon is

3. 818 × 105 km from the earth and find the distance

from the center of the earth to the center of mass of the

moon and the earth.

b.Find the reduced mass of the moon and the earth and

express it as a percentage of the mass of the

moon.

17.2 The Relative Schrödinger Equation.

Angular Momentum

The relative Schrödinger equation cannot be solved in Cartesian coordinates. We trans-

form to spherical polar coordinates in order to have an expression for the potential

energy that contains only one coordinate. Spherical polar coordinates are depicted in

Figure 17.3. The expression for the Laplacian operator in spherical polar coordinates

is found in Eq. (B-47) of Appendix B. The relative Schrödinger equation is now

Ĥrelψ−h ̄

2

2 μr^2

[

∂

∂r

(

r^2

∂ψ

∂r

)

+

1

sin(θ)

∂

∂θ

(

sin(θ)

∂ψ

∂θ

)

+

1

sin^2 (θ)

∂^2 ψ

∂φ^2

]

+V(r)ψ

Erelψ (17.2-1)